Equivalence of viscosity and weak solutions for a $p$-parabolic equation

We study the relationship of viscosity and weak solutions to the equation \[ \smash{\partial_{t}u-\Delta_{p}u=f(Du)} \] where $p>1$ and $f\in C(\mathbb{R}^{N})$ satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $p\geq2$.


Introduction
A classical solution to a partial differential equation is a smooth function that satisfies the equation pointwise. Since many equations that appear in applications admit no such solutions, a more general class of solutions is needed. One such class is the extensively studied distributional weak solutions defined by integration by parts. Another is the celebrated viscosity solutions based on generalized pointwise derivatives. When both classes of solutions can be meaningfully defined, it is naturally crucial that they coincide. This has been profusely studied starting from [Ish95]. In [JLM01] the equivalence of solutions was proved for the parabolic p-Laplacian. The objective of the present work is to prove this equivalence in a different way while also allowing the equation to depend on a first-order term. To the best of our knowledge, the proof is new even in the homogeneous case, at least when 1 < p < 2.
More precisely, we study the parabolic equation where 1 < p < ∞ and f ∈ C(R N ) satisfies a certain growth condition, for details see Section 2. We show that bounded viscosity supersolutions to (1.1) coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions in the range p ≥ 2 under slightly stronger assumptions on f. To show that viscosity supersolutions are weak supersolutions, we apply the technique introduced by Julin and Juutinen [JJ12]. In contrast to [JLM01], we do not employ the uniqueness machinery of viscosity solutions. Instead, our strategy is to approximate a viscosity supersolution u by its inf-convolution u ε . It is straightforward to show that u ε is still a viscosity supersolution in a smaller domain. This and the pointwise properties of the inf-convolution imply that u ε is also a weak supersolution in the smaller domain. Furthermore, it follows from Caccioppoli's estimates that u ε converges to u in a suitable Sobolev space. It then remains to pass to the limit to see that u is a weak supersolution.
To show that weak supersolutions are viscosity supersolutions, we apply the argument from [JLM01] that is based on the comparison principle of weak solutions. However, we could not find a reference for comparison principle for the equation (1.1). Therefore we give a detailed proof of such a result.
To prove the lower semicontinuity of weak supersolutions, we adapt the strategy of [Kuu09]. First we prove estimates for the essential supremum of a subsolution using the Moser's iteration technique. Then we use those estimates to deduce that a supersolution is lower semicontinuous at its Lebesgue points.
The equivalence of viscosity and weak solutions for the p-Laplace equation and its parabolic version was first proven in [JLM01]. A different proof in the elliptic case was found in [JJ12]. Recently the equivalence of solutions has been studied for various equations. These include the normalized p-Poisson equation [APR17], a non-homogeneous p-Laplace equation [MO] and the normalized p(x)-Laplace equation [Sil18]. Moreover, in [PV] the equivalence is shown for the radial solutions of a parabolic equation. We also mention that an unpublished version of [Lin12] applies [JJ12] to sketch the equivalence of solutions to (1.1) in the homogeneous case when p ≥ 2.
Comparison principles for quasilinear parabolic equations have been studied by several authors. In [Jun93] comparison is proven for ∂ t u − ∆ p u + f (u, x, t) = 0 when p > 2 and f is a continuous function such that |f (u, x, t)| ≤ g(u) for some g ∈ C 1 . The homogeneous case for the p-parabolic equation is considered also in [?] and the general equation ∂ t u − div A(x, t, Du) = 0 in [KKP10]. Equations with gradient terms are studied for example in [Att12], where comparison principle is shown for the equation ∂ t u − ∆ p u − |Du| β = 0 when p > 2 and β > p − 1. In the recent papers [BT14,BT], both positive results and counter examples are provided for the comparison, strong comparison and maximum principles for the equation ∂ t u − ∆ p u − λ |u| p−2 u − f (x, t) = 0. Furthermore, according to [BGKT16], the equation ∂ t u − ∆ p u = q(x) |u| α can admit multiple solutions with zero boundary values when 0 < α < 1.
The paper is organized as follows. Section 2 contains the precise definitions of weak and viscosity solutions. In Section 3 we show that weak supersolutions are viscosity supersolutions, and the converse is shown in Section 4. Finally, the lower semicontinuity of weak supersolutions is considered in Section 5.

Preliminaries
The symbols Ξ and Ω are reserved for bounded domains in R N × R and R N , respectively. For t 1 < t 2 , we define the cylinder Ω t 1 ,t 2 := Ω × (t 1 , t 2 ) and its parabolic boundary The Sobolev space W 1,p (Ω) contains the functions u ∈ L p (Ω) for which the distributional gradient Du exists and belongs in L p (Ω). It is equipped with the norm A Lebesgue measurable function u : Ω t 1 ,t 2 → R belongs to the parabolic Sobolev space L p (t 1 , t 2 ; W 1,p (Ω)) if u(·, t) ∈ W 1,p (Ω) for almost every t ∈ (t 1 , t 2 ) and the norm Ωt 1 ,t 2 |u| p + |Du| p dz 1 p is finite. By dz we mean integration with respect to space and time variables, i.e. dz = dx dt. Integral average is denoted by Growth condition. Unless otherwise stated, the function f ∈ C(R N ) is assumed to satisfy the growth condition where C f > 0 and 1 ≤ β < p.
for all non-negative test functions ϕ ∈ C ∞ 0 (Ω t 1 ,t 2 ). For weak subsolutions the inequality is reversed and a function is a weak solution if it is both super-and subsolution.
To define viscosity solutions to (1.1), we set for all ϕ ∈ C 2 with Dϕ = 0

Definition 2.2 (Viscosity solution). A lower semicontinuous and bounded function
An upper semicontinuous and bounded function u : Ξ → R is a viscosity subsolution to (1.1) in Ξ if whenever ϕ ∈ C 2 (Ξ) and (x 0 , t 0 ) ∈ Ξ are such that A function that is both viscosity sub-and supersolution is a viscosity solution.
If a function ϕ is like in the definition of viscosity supersolution, we say that ϕ touches u from below at (x 0 , t 0 ). The limit supremum in the definition is needed because the operator ∆ p is singular when 1 < p < 2. When p ≥ 2, the operator is degenerate and the limit supremum disappears.

Weak solutions are viscosity solutions
We show that bounded, lower semicontinuous weak supersolutions to (1.1) are viscosity supersolutions when 1 < p < ∞ and f ∈ C(R N ) satisfies the growth condition (G1). One way to prove this kind of results is by applying the comparison principle [JLM01]. However, we could not find the comparison principle for the equation (1.1) in the literature and therefore we prove it first. To this end, we first prove comparison Lemmas 3.2 and 3.3 for locally Lipschitz continuous f. The local Lipschitz continuity allows us to absorb the first-order terms into the terms that appear due to the p-Laplacian, see Step 2 in proof of Lemma 3.2. To deal with general f, we take a locally Lipschitz continuous approximant f δ such that f − f δ L ∞ (R N ) < δ/4T . Then for sub-and supersolutions u and v, we consider the functions and v δ :  [Jun93]. See also Chapters 3.5 and 3.6 in [PS07] for the elliptic case. A minor difference in our results is that instead of requiring that both the subsolution and the supersolution have uniformly bounded gradients, we only require this for the subsolution.
To prove the comparison principle, we need to use a test function that depends on the supersolution itself. However, supersolutions do not necessarily have a time derivative. One way to deal with this is to use mollifications in the time direction. For a compactly supported ϕ ∈ L p (Ω T ) we define its time-mollification by where ρ ǫ is a standard mollifier whose support is contained in (−ǫ, ǫ). Then ϕ ǫ has time derivative and ϕ ǫ → ϕ in L p (Ω T ). Furthermore, the time-mollification of a supersolution satisfies a reguralized equation in the sense of the following lemma.
If ϕ is smooth, then testing the weak formulation of (1.1) with ϕ ǫ , changing variables and using Fubini's theorem yields (3.1). The general case follows by approximating ϕ in W 1,p (Ω T ) with the standard mollification. We omit the details.
Lemma 3.2. Let 1 < p < 2 and let f be locally Lipschitz. Let u, v ∈ L ∞ (Ω T ) respectively be weak sub-and supersolutions to (1.1) in Ω T . Assume that for all Suppose also that Du ∈ L ∞ (Ω T ). Then u ≤ v a.e. in Ω T .
Proof. (Step 1) Let l > 0 and set w := (u − v − l) + . Let also s ∈ (0, T ). We want to use w · χ [0,s] as a test function, but since it is not smooth, we must perform mollifications. Let h > 0 and define The function ϕ is compactly supported and belongs in W 1,p (Ω T ). Therefore by Lemma 3.1 we have We use the linearity of convolution and integration by parts to eliminate the time derivative. We obtain Moreover, by the Lebesgue differentiation theorem for a.e. s ∈ (0, T ) it holds The terms at the right-hand side of (3.2) converge similarly. Hence for a.e. s ∈ (0, T ) we have Step 2) We seek to absorb some of I 1 into I 2 so that we can conclude from Grönwall's inequality that w ≡ 0 almost everywhere. Since f is locally Lipschitz continuous, there are constants M ≥ max(2 Du L ∞ (Ω T ) , 1) and L = L(M) such that (3.4) We denote Ω + s := {x ∈ Ω s : w ≥ 0}, A := Ω + s ∩ {|Dv| < M} and B := Ω + s ∩ {|Dv| ≥ M} . Observe that in B we have by the growth condition (G1), choice of M and the assump- (3.6) It follows from (3.4), (3.5), (3.6) and Young's inequality that where in the last step we used that p p−β > 2 to estimate Using the vector inequality which holds when 1 < p < 2 [Lin17, p98], we get where C(p, M), C(p) > 0. Combining the estimates (3.7) and (3.9) we arrive at 3) and taking small enough ǫ yields Since this holds for a.e. s ∈ (0, T ), Grönwall's inequality implies that w ≡ 0 a.e. in Ω T . Finally, letting l → 0 yields that u − v ≤ 0 a.e. in Ω T .
Proof. Let l > 0 and set w : Repeating the first step of the proof of Lemma 3.2, we arrive at the inequality Using the vector inequality which holds when p ≥ 2 [Lin17, p95], we get we arrive at Combining (3.11) and (3.13) with (3.10) we get By taking small enough ǫ = ǫ(p), the above becomes (3.14) Observe that since w is bounded and p p−1 , p p−β > 1, the integrand at the right-hand side is bounded by some constant times w 2 . To argue this rigorously, we write down the following algebraic fact.
Next we use the previous comparison results to prove the comparison principle for general continuous f .
Proof. For δ > 0, define Then for any non-negative test function ϕ ∈ C ∞ 0 (Ω T ) we have by integration by parts g. [Mic00]). Then, since u is a weak subsolution, we have Hence u δ is a weak subsolution to Now it follows from the comparison Lemmas 3.2 and 3.3 that u δ ≤ v δ a.e. in Ω T . Thus a.e. in Ω T .
Now that the comparison principle is proven, we are ready to show that weak solutions are viscosity solutions.
Proof. Assume on the contrary that there is φ ∈ C 2 (Ξ) touching u from below at It follows from above that there are r > 0 and δ > 0 such that Indeed, otherwise there would be a sequence (x n , t n ) → (x 0 , t 0 ) such that x n = x 0 and but this contradicts (3.15). Using Gauss's theorem and (3.16) we obtain for any non- Let l := min ∂pQr (u − φ) > 0 and set φ := φ + l. Then by the above inequality, φ is a weak subsolution to

Viscosity solutions are weak solutions
We show that bounded viscosity supersolutions to (1.1) are weak supersolutions when 1 < p < ∞ and f ∈ C(R N ) satisfies the growth condition (G1). We use the method developed in [JJ12]. The method of [JJ12] was previously applied to parabolic equations in [PV], but for radially symmetric solutions.
The idea is to approximate a viscosity supersolution u to (1.1) by the inf-convolution where ε > 0 and q ≥ 2 is a fixed constant so large that p − 2 + q−2 q−1 > 0. It is straightforward to show that the inf-convolution u ε is a viscosity supersolution in the smaller domain where r(ε), t(ε) → 0 as ε → 0. Moreover, u ε is semi-concave by definition and therefore it has a second derivative almost everywhere. It follows from these pointwise properties that u ε is a weak supersolution to (1.1) in Ξ ε . Caccioppoli type estimates then imply that u ε converges to u in a parabolic Sobolev space and consequently u is a weak supersolution.
The standard properties of the inf-convolution are postponed to the end of this section. Instead, we begin by proving the key observation: that the inf-convolution of a viscosity supersolution is a weak supersolution in the smaller domain Ξ ε . When p ≥ 2, the idea is the following. Since u ε is a viscosity supersolution to (1.1) that is twice differentiable almost everywhere, it satisfies the equation pointwise almost everywhere. Hence we may multiply the equation by a non-negative test function ϕ and integrate over Ξ ε so that the integral will be non-negative. Then we approximate this expression through smooth functions u ε,j defined via the standard mollification. Since u ε,j is smooth, we may integrate by parts to reach the weak formulation of the equation, see (4.1). It then remains to let j → ∞ to conclude that u ε is a weak supersolution. The range 1 < p < 2 is more delicate because of the singularity of the p-Laplace operator and therefore we consider the case p ≥ 2 first.
Proof. Fix a non-negative test function ϕ ∈ C ∞ 0 (Ξ ε ). By Remark 4.8, the function is concave in Ξ ε and we can approximate it by smooth concave functions φ j so that Since u ε,j is smooth and ϕ is compactly supported in Ξ ε , we integrate by parts to get We intend to use Fatou's lemma at the left-hand side and dominated convergence at the right-hand side. Once we verify their assumptions, we arrive at the inequality The left-hand side is non-negative since by Lemma 4.7 the inf-convolution u ε is still a viscosity supersolution in Ξ ε . Consequently u ε is a weak supersolution in Ξ ε as desired.
It remains to justify our use of Fatou's lemma and the dominated convergence theorem. It follows from Remark 4.8 that |u ε,j |, |∂ t u ε,j | and |Du ε,j | are uniformly bounded by some constant M > 0 in the support of ϕ with respect to j. This justifies our use of the dominated convergence theorem. Observe then that since φ j is concave, we have The integrand at the left-hand side of (4.1) is therefore bounded from below with respect to j, justifying our use of Fatou's lemma.
Next we consider the singular case 1 < p < 2. We cannot directly repeat the previous proof because ∆ p u ε no longer has a clear meaning at the points where Du ε = 0. To deal with this, we consider the regularized terms where ∆ ∞ u = D 2 uDu, Du .
where C(q, ε, u) is the semi-concavity constant of u ε in Ξ ε . Then by Remark 4.8 we can approximate φ by smooth concave functions φ j so that Let δ ∈ (0, 1). Since u ε,j is smooth and ϕ is compactly supported in Ξ ε , we calculate via integration by parts Recalling the shorthand ∆ p,δ defined in (4.2), we deduce from the above that We use Fatou's lemma at the left-hand side and the dominated convergence at the right-hand side. Once we verify their assumptions, we arrive at the auxiliary inequality (4.5) Next we verify the assumptions of Fatou's lemma and the dominated convergence theorem. By Remark 4.8, the functions |u ε,j |, |∂ t u ε,j | and |Du ε,j | are uniformly bounded by some constant M > 1 in the support of ϕ with respect to j. Hence the assumptions of the dominated convergence theorem are satisfied. Observe then that the concavity of φ j implies that D 2 u ε,j ≤ C(q, ε, u)I. Thus the integrand at the left-hand side of (4.4) has a lower bound independent of j when Du ε,j = 0. When Du ε,j = 0, we have so that our use of Fatou's lemma is justified. ( Step 2) We let δ → 0 in the auxiliary inequality (4.5). Since u ε is Lipschitz continuous, the dominated convergence theorem implies Applying Fatou's lemma (we verify assumptions at the end), we get where the last inequality follows from Lemma 4.7 since u ε is twice differentiable almost everywhere. Combining (4.6) and (4.7), we find that u ε is a weak supersolution in Ξ ε . It remains to verify the assumptions of Fatou's lemma, i.e. that the integrand at the left-hand side of (4.6) has a lower bound independent of δ. When Du ε = 0, this follows directly from the inequality which holds by Lemma 4.6. When Du ε = 0, we recall that by Lipschitz continuity ∂ t u ε and Du ε are uniformly bounded in Ξ ε , and estimate where we used that p − 2 + q−2 q−1 > 0. Hence the assumptions of Fatou's lemma hold.
If u ε is the sequence of inf-convolutions of a viscosity supersolution to (1.1), then by next Caccioppoli's inequality the sequence Du ε converges weakly in L p loc (Ξ) up to a subsequence. However, we need stronger convergence to pass to the limit under the integral sign of For this end, we show in Lemma 4.4 that Du ε converges in L r loc (Ξ) for all 1 < r < p.

Lemma 4.3 (Caccioppoli's inequality). Let 1 < p < ∞. Assume that u is a locally Lipschitz continuous weak supersolution to (1.1) in Ξ. Then there is a constant
Proof. Since u is locally Lipschitz continuous, the function ϕ := (M − u) ξ p is an admissible test function. Testing the weak formulation of (1.1) with ϕ yields (4.8)

We have by integration by parts
Using the growth condition (G1) and Young's inequality we get Combining these estimates with (4.8) and absorbing the terms with Du to the left-hand side yields the desired inequality.
The proof of Lemma 4.4 is based on that of Lemma 5 in [LM07], see also Theorem 5.3 in [KKP10]. For the convenience of the reader, we give the full details.
Lemma 4.4. Let 1 < p < ∞. Suppose that (u j ) is a sequence of locally Lipschitz continuous weak supersolutions to (1.1) such that u j → u locally uniformly in Ξ. Then (Du j ) is a Cauchy sequence in L r loc (Ξ) for any 1 < r < p. Proof. Let U ⋐ Ξ and take a cut-off function θ ∈ C ∞ 0 (Ξ) such that 0 ≤ θ ≤ 1 and θ ≡ 1 in U. For δ > 0, we set Then the function (δ−w jk )θ is an admissible test function with a time derivative since it is Lipschitz continuous. Since u j is a weak supersolution, testing the weak formulation of (1.1) with (δ − w jk )θ yields Since |w jk | ≤ δ and Dw jk = χ {|u j −u k |<δ} (Du j − Du k ), the above becomes Since u k is a weak supersolution, the same arguments as above but testing this time with (δ + w jk )θ yield the analogous estimate Summing up these two inequalities we arrive at (4.9) We proceed to estimate these integrals. Denoting M := sup j u j L ∞ (spt θ) < ∞, we have by the Caccioppoli's inequality Lemma 4.3 (4.10) The estimate (4.10) and Hölder's inequality imply that To estimate I 2 , we also use the growth condition (G1) and the assumption β < p. We get The integral I 3 is estimated using integration by parts and that |w jk | ≤ δ For the last integral we have directly I 4 ≤ δC(θ, M). Combining these estimates with (4.9) we arrive at where C 0 = C(p, β, C f , θ, M). If 1 < p < 2, Hölder's inequality and the algebraic inequality (3.8) give the estimate (recall that 1 < r < p and θ ≡ 1 in U) where in the last inequality we also used (4.10) with the knowledge r(2−p) (2−r) ≤ p(2−p) 2−p = p. If p ≥ 2, Hölder's inequality and the algebraic inequality (3.12) imply Hence (4.11) leads to On the other hand, Hölder's and Tchebysheff's inequalities with (4.10) imply β, r, C f , θ, M). Taking first small δ > 0 and then large j, k, we can make the right-hand side arbitrarily small. Now we are ready to prove the main result of this section which states that bounded viscosity supersolutions are weak supersolutions. Proof. Fix a non-negative test function ϕ ∈ C ∞ 0 (Ξ) and take an open cylinder Ω t 1 ,t 2 ⋐ Ξ such that spt ϕ ⋐ Ω t 1 ,t 2 . Let ε > 0 be so small that Ω t 1 ,t 2 ⋐ Ξ ε . Then Lemma 4.2 implies that u ε is a weak supersolution to (1.1) in Ξ ε . Therefore by the Caccioppoli's inequality Lemma 4.3, Du ε is bounded in L p (Ω t 1 ,t 2 ). Hence Du ε converges weakly in L p (Ω t 1 ,t 2 ) up to a subsequence. Since also u ε → u in L ∞ (Ω t 1 ,t 2 ), it follows that u ∈ L p (t 1 , t 2 ; W 1,p (Ω)).
On the other hand, we have |f (Du ε ) − f (Du)| → 0 a.e. in Ω t 1 ,t 2 up to a subsequence and the integrand in I 1 is dominated by an integrable function since the growth condition (G1) implies Hence, for any M ≥ 1, we have I 1 → 0 as ε → 0 by the dominated convergence theorem. By taking first large M ≥ 1 and then small ε > 0, we can make I 1 + I 2 arbitrarily small.
The rest of this section is devoted to the properties of the inf-convolution. The facts in the following lemma are well known, see e.g. [CIL92], [JJ12], [Kat15] or [PV].
Lemma 4.6. Assume that u : Ξ → R is lower semicontinuous and bounded. Then u ε has the following properties.
(i) We have u ε ≤ u in Ξ and u ε → u locally uniformly as ε → 0.
(ii) Denote r(ε) := (qε q−1 osc Ξ u) Then for any ( (iii) The function u ε is semi-concave in Ξ ε with a semi-concavity constant depending only on u, q and ε. (iv) Assume that u ε is differentiable in time and twice differentiable in space at (x, t) ∈ Ξ ε . Then Next we show that the inf-convolution of a viscosity supersolution to (1.1) is still a supersolution in the smaller domain Ξ ε . Since the inf-convolution is "flat enough", that is, since q > p/(p − 1), the inf-convolution essentially cancels the singularity of the p-Laplace operator. This allows us to extract information on the time derivative at those points of differentiability where Du ε vanishes.
Proof. Assume that ϕ touches u ε from below at (x, t) ∈ Ξ ε . Let (x ε , t ε ) be like in the property (ii) of Lemma 4.6. Then Then ψ touches u from below at (x ε , t ε ) since by (4.14) and selecting (y, τ ) = (z + x − x ε , s + t − t ε ) in (4.15) gives Since u is a viscosity supersolution, it follows that (Dϕ(z, s))) , and the first claim is proven. To prove the second claim, assume that u ε is differentiable in time and twice differentiable in space at (x, t) ∈ Ξ ε and Du ε (x, t) = 0. By the property (iv) in Lemma 4.6, we have x = x ε , so that Hence by the definition of inf-convolution for all (y, s) ∈ Ξ.
Arranging the terms as we see that the function φ touches u from below at (x, t ε ). Since u is a viscosity supersolution and Dφ(y, s) = 0 when y = x, we have lim sup On the other hand, since q > p/(p − 1), we have ∆ p φ(y, s) → 0 as y → x. Hence we get where the last equality follows from the property (iv) in Lemma 4.6.

Lower semicontinuity of supersolutions
We show the lower semicontinuity of weak supersolutions when p ≥ 2 and the function f ∈ C(R N ) satisfies that f (0) = 0 as well as the stronger growth condition Our proof follows the method of Kuusi [Kuu09], but the first-order term causes some modifications. In particular, our essential supremum estimate is slightly different, see Theorem 5.3 and the brief discussion before it. The assumption f (0) = 0 is used to ensure that the positive part u + of a subsolution is still a subsolution.
We begin by proving estimates for the essential supremum of a subsolution using the Moser's iteration technique. We first need the following Caccioppoli's inequalities.
Lemma 5.1 (Caccioppoli's inequalities). Assume that p ≥ 2 and that (G2) holds. Suppose that u is a non-negative weak subsolution to (1.1) in Ω t 1 ,t 2 and u ∈ L p−1+λ (Ω t 1 ,t 2 ) for some λ ≥ 1. Then there exists a constant C = C(p, C f ) that satisfies the estimates ess sup Proof. We test the regularized equation in Lemma (3.1) with ϕ := min(u ǫ , k) λ−1 u ǫ ζ p η, where η is the following cut-off function and t 1 < s < τ < t 2 , h > 0. We denote g(l) := l 0 min(r, k) λ−1 r dr. Then integration by parts and Lebesgue's differentiation theorem yield for a.e. s, τ ∈ (t 1 , t 2 ) Letting s → t 1 and observing that the other terms of (3.1) converge as well, we obtain for a.e. τ ∈ (t 1 , t 2 ) that where we have denoted u k := min(u, k). Since we have by Young's inequality Moreover, by the growth condition (G2) and Young's inequality Collecting the estimates, moving the terms with Du to the left-hand side and letting k → ∞, we arrive at Since the integrals are positive, this yields the first inequality of the lemma by taking essential supremum over τ . The second inequality follows from (5.1) by using that We first prove the following essential supremum estimate where we assume that the subsolution is bounded away from zero.
Assuming that we already know that u ∈ L p−1+λ j (U j ), then we have by a parabolic Sobolev's inequality (see [DiB93,p7] The first estimate in Lemma 5.1 gives Using the second estimate with ζ = ϕ Combining (5.3) with (5.4) and (5.5) we arrive at where γ = 1 + p/N. We wish to iterate this inequality, but having multiple terms at the right-hand side is a problem. This is where the assumption (5.2) comes into play. Since u ≥ (R p /T ) 1/(p−1) , we have and since T /R p ≥ 1, we have also Using these estimates it follows from (5.6) that Observe that κα = p − 1 + λ j (1 + p/N) + p/N = p − 1 + λ j+1 .
Next we consider the case where the non-negative subsolution is not necessarily bounded away from zero. Observe that the estimate differs from the usual estimate for the p-Laplacian because of the power 1/(p − 1) in the first term (cf. [DiB93, Theorem 4.1] or [Kuu09, Theorem 3.4]). However, we have the additional assumption (5.10). Theorem 5.3. Assume that p ≥ 2 and that (G2) holds. Suppose that u is a nonnegative weak subsolution to (1.1) in Ξ and B R (x 0 ) × (t 0 − T, t 0 ) ⋐ Ξ with R, T < 1 such that R p T ≤ 1. (5.10) Then there exists a constant C = C(N, p, C f , δ) such that we have the estimate ess sup Proof. We denote Using Lemma 5.2 on the subsolution v := θ + u we get the estimate ess sup Taking σ = 1/2 now yields the desired inequality.
Then u = u * almost everywhere.
Proof. For all M ∈ N, we define the cylinders Q M R (x, t) := B R (x) × (t − MR p , t + MR p ). We denote by E M the set of Lebesgue points with respect to the basis {Q M R }, that is, |u(x, t) − u(y, s)| p− 1 2 dy ds = 0 .
Let R 0 be a radius such that ess lim inf (x,t)→(x 0 ,t 0 ) u(x, t) − ess inf We set v := (u(x 0 , t 0 ) − u) + . Since (x 0 , t 0 ) ∈ E, we find for any M ∈ N a radius R 1 = R 1 (M) such that (5.13) On the other hand, by Lemma 5.4 the function v is a weak subsolution to where g(ξ) = −f (−ξ). Observe also that the cylinder Q M R 1 (x 0 , t 0 ) satisfies the condition (5.10) since R p 1 /(MR p 1 ) ≤ 1. Hence we may apply Theorem 5.3 with δ = 3/2 and then